How to verify classification model trained on classification dataset on a detection dataset for classification purpose?
To verify the accuracy of the classification stage, you will need labeled images with a single car.
To train and verify accuracy of the detection stage and full system, you can:

in the datasets with images with multiple cars, manually, mark the image rectangles that contains one car.
from previous, split the image in one or more ones, each one containing a single car.
pass each one of the previous image with a single car to the classification stage (that means assume classification has 100% accuracy). Record its outputs (labeled cars).
now, from output of steps 1) and 3), you can produce labeled images with multiple cars. Use it to train the detector and verify full system accuracy.